image/svg+xml Electron 1 and 2kinetic energy Electron 1 and 2attraction to nucleus Electron –electron repulsionThis is the problem term! 2-electronwavefunction Potential energy surfaces All computational chemistry methods simplify the potential energy surface in some wayThis requires chemical knowledge about the method and the potential energy surface Resolution Electrons essentially behave as standing waves in 3D space Node (Zero amplitude) Anti-nodes (Maximum amplitude) Increasing nodes and frequency (energy) Planck–Einstein relation relates frequency to energy: E = hf More nodes = higher-energy standing wave 0 Nodes + +–1 Node 2 Nodes ++–– 0 Nodes1 Node2 Nodes3 Nodesspdf Increasing nodes and frequency (energy) The calculations include various properties important to much of chemistry:• Electronic/molecular orbitals• Energy quantisation• Dispersion interactions• Exchange and correlation energy • Generally much more accurate than classical methods, but take much longer• Usually used for calculating small systems No such thing as an absolute energy. All energies are relative in chemistry • In computational chemistry the energy depends on the method used• Key point: only energy differences typically matter ΔE = +25 kJ mol-1 What are systems are classical methods good for? • Biological systems• DNA, proteins etc. (many atoms)• Membranes (cannot consider isolated molecules)• Materials• Pure liquids, ionic liquids, polymers, liquid crystals (cannot consider isolated molecules, properties must be time-averaged)• Bulk properties• Density, diffusion, viscosity, phase changes (arise from molecular ensembles, properties change over significant time-scales)• All these systems typically require thousands of atoms (or more) to describe them well• These can be considered “large” systems • Many (most) chemical systems are not in their lowest energy states under standard conditions • Must consider only classical interactions• I.e. no wave-behaviour of particles• All particles treated classically i.e. as we treat macroscopic particles• Electrons only well described by quantum methods, so classical methodslimited to atoms as the smallest particles • To treat atoms classically we must make some (big!) approximations U r AB small k large k U θ ABC small k large k ABCDϕ 0 30 60 90 120 150 180 210 240 270 300 330 360 U ϕ simulated bonds behave like classical springs• Small k = bond easily stretched (e.g. C-O, O-H)• Large k = bond hard to stretch (e.g. C=C, C≡C) • Small kABC = bond easily bent, e.g Si-O-Si• Large kABC = bond angle varies little, e.g. Torsion potentials describe ease of bond rotation• bonding characteristics (e.g. σ and π characteristics)• E.g. ethane H-C-C-H is staggered ethene H-C=C-H is eclipsed Electrostatic interactions can be described using Coulomb’s law ij r U r Like chargesrepel U r Opposite chargesattract Enables simulation of dipole-dipole interactions and H-bonding Describe van der Waals interactions using Lennard-Jones potential • Repulsive at short distance (electron-electron repulsion)• Attractive at longer distance U r Classical force fields• Data files contain parameters to define many types of molecule• Definition of atom types underpins each force-field• Cannot simply define parameters for each element: parameters can depend on environment• Force field files are typically thousands of lines long• Each parameter must be carefully defined • Force-fields vary between being very specific and very general• Specific force-fields tend to be very good at simulating what they are designed for, but are very poor for other systems (lack transferability)• General force-fields tend to perform reasonably well for a wide range of systems (transferable) • Complex systems we are typically interested in have significant kinetic energy (e.g. room temperature)• Kinetic energy is defined by the temperature, T, of a system• We myst consider many different configurations• A big challenge is simulating representative configurations • The energy of a configuration is related to its probability:• Low energy = high probability• High energy = low probability Classical Methods 2 Low energy, high probability High energy, low probability • In chemistry we typically observe a lot of molecules• Therefore for chemical observations, the most likely configuration is an excellent representation of what we observe• What does this mean for us? In computational chemistry, we only need to calculate the most likely configuration Energy Relative population25 %50 %15 %10 %This is a configuration,i.e. a set of occupancies 8 half-chairboattwist-boatchair Energy 25 %50 %15 %10 %Possible ratios in solutionThis is a configuration, i.e. a setof occupancies/populations ofdifferent energy levels 25 %25 %25 %25 % This is an alternative configuration A chemical equivalent could be cyclohexane: Statistical thermodynamics enables us to relate energy levels to populations (and therefore chemical properties) 𝑃𝐸𝑖∝𝑒−𝐸𝑖𝑘𝑇 Probability of a state being populated Energy of the state Constant ×temperature 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Energy level Relative population T5T10T100T Gives quantitative prediction of most likely configuration from temperature and energy levels Molecular dynamics • Molecular dynamics is a method by which the (many dimensional) potentialenergy landscape may be explored• Give system certain amount of energy overall (thermostat)• Let it do what it wants over time, i.e. let the system explore the PES Basic method• Randomly sample the PES• In random sampling, high energy configurations may be as likely as low energy ones; equivalent to having infinite temperature• Weight different configurations based on their energy (lower energy; larger weighting) • Definition of a force field means that the system energy is known for any set of atomic/molecular positions. • Aims to match “real” conditions• Input temperature/pressure/volume• Run simulation for finite time• Takes advantage of the fact that each configuration determines the next • Big instantaneous jumps in atomic/molecular positions are not possible, just as they aren’t in reality• MD trajectory maps a path through phase space (the potential energy surface) 9 10 Region exploredRegionunexploredRegionunexploredStartingpoint • Temperature defines maximum energy of a system• Energy barriers may be too high for given temperature to cross• Other (important) low energy configurations may not be explored• This can be a severe limitation of molecular dynamics ≠ Time averageStatistical averageStartingpoint Ergodic hypothesis• Essentially states that a time average is the same as a statistical average • i.e. that an MD simulation will explore all regions that should be explored according to the potential energy surface• This is a fundamental assumption of MD ! Multiple starting configurations can overcome energy barrier problems MD simulation 1 MD simulation 2MD simulation 3 Starting point Starting pointStarting point Statistical average Equilibration can also be a problem with MD: The starting geometry may not be a good representation of the final geometry Simulation time Region for analysis Equilibration • Start with “reasonable” system configuration• Calculate energy• Randomly perturb system to give new configuration• Calculate new energy - If new energy is lower than old energy: • accept the perturbation - If new energy is higher than old energy • Generate random number between 0 and 1 - If Boltzmann probability of energy change is greater than the random number, accept the perturbation - If Boltzmann probability of energy change is less than the random number, reject the perturbation• Repeat• What do we gain? Very high energy states unlikely to be sampled• Not wasting computer power sampling unwanted regions of the PES Metropolis Monte Carlo - a refined approach • In practice, perturbation step usually involves randomly translating androtating 1 molecule, e.g. in a liquid MD:• Time evolution matches experimental conditions• Cannot readily overcome energy barriers• Can simulate time-dependent processes (e.g. diffusion, docking) MC:• Random selection of states does not match experiment• Energy barriers easily overcome (with correct perturbations)• Cannot simulate time-dependent processes Atomic structure How has the current view of atomic structure come about? What lead to current quantum theory? Experiment 1: scattering of alpha-particles by gold foil Results:• Most particles went straight through• Some were deflected• A few were reflected backConclusions:• Atoms are mainly empty space• Positive charges must be concentrated in small regions Experiment 2: Hydrogen emission spectrum • Apply high voltage to H2 tube and measure emitted light• Voltage electronically excites electrons• Light emitted as they relax back to their ground stateResults: only specific wavelengths observed • Conclusion: electrons only take specific energies Experiment 3: X-ray/electron diffraction X-ray/electronsourceX-ray/electronbeamSampleDiffractedbeamDetector LightElectrons Conclusion: particles and waves can behave in the same way Describing atoms as small positive charges (nuclei) surrounded by wave-like negative charges(electrons) explains these observations:- Diffuse, wave-like electrons do not interact strongly with high-energy alpha particles- Wave-like electrons explains quantisation- Wave-like electrons explains diffraction n = 3n = 4n = 5n = 6 Fundamental concept of quantum mechanics is that there is a wavefunction,Ψ, that exists for any (chemical) system (i.e. particles behave as waves)Born-Oppenheimer approximation: the assumption that the motion of atomic nucleiand electrons in a molecule can be separated• Nuclear motion is much slower than electron motion• Consider nuclei to be static; just calculate for electronsThe wavefunction of a chemical system is the wave-description of theelectrons within the systemThe goal of quantum methods in computational chemistry is to find the fullwavefunction of the system, Ψ(x,t)A wavefunction can be imagined as set of standing waves that (together)define the system Born interpretation: Ψ2 = probability of finding the particle anywhere in space Shaded region must add up to 1 x Function is infiniteNot acceptable as awavefunction x The probability of finding the particle at x must be meaningful For any value of x, there must only be one value of ψ(x) Function has two yvalues atx= 5Not acceptable as a wavefunction HΨ = EΨ The Schrodinger equation enables determination of the energyof a system from the wavefunction Representing a multi-electron wavefunction How do we determine repulsion between wave-like electrons? The orbital approximation Assume that a reasonable approximation for a many-electronwavefunction is to consider each electron occupying its own “hydrogenic” orbital 1 two-electronwavefunction 2 one-electronwavefunctions Conceptually this is OK, but we also need to think about electron spin Spinorbital just a fancy way of defining an orbital + electron spin, e.g. 1sαspinorbital:n = 1l= 0ml= 0ms= ½dz2βspinorbital:n = 3l= 2ml= 0ms=-½ We must compine 1-electron spinorbitals in such a way that:- No 2 electrons of the same spin can be in the same orbital (Pauli exclusion principle)- Electrons are indistinguishable, i.e. equally associated with each spinorbital This combination satisifies these requirements A mathematical shorthand way of writing this expression is as a Slater determinant: In general for N electrons we can write this sort of expression as: Spinorbitals Electrons 𝑒 2 4 π ε 0 𝑟 12 • E.g. H2 molecule: the potential energy of each 1-electron orbital (spinorbital) depends on the other • Need to approximate e- e- repulsion term in our Hamiltonian operator so it depends only on one electron Solving this is like trying to solve 1 equation with 2 unknowns Douglas Hartree and Vladimir Fock replaced the e- e- repulsion term in the Schrodinger equation with such an approximation This e- e- repulsion term accounts for potential energyin average field created by all other electrons Final barrier to finding a solution: J and K depend of the wavefunction. Key is to use the variational principle The exact wavefunction will correspond to the one with the lowest ground state energy, E0, e.g Poor guesseshigh energyGood guessLow energy E0 gives us a measure of how good our approximation is:lower E0 = better approximation The best wavefunction, ψ, is the one which gives us the lowest E0 • Guess at orbitals - E.g. pick arbitrary parameters to describe hydrogenic orbitals• Construct Fock operator (J & K) from these orbitals - This means we now have an operator and orbitals• Solve the equation using operator and guessed orbitals - Produces new orbitals and new (lower) energy• Construct a new Fock operator• Repeat!• Keep going until the input orbitals match the output orbitals• Solution is self-consistent: output ends up the same as input Self-consistent field (SCF) method HF method accounts for ca. 99% of total energy1% is due to assumption that Slater determinant is sufficient to describe wavefunction i.e. Electron-electron interactions (correlation) are averaged; assumes electrons move independently • In reality because electrons interact and repel each other, their motion in atoms is correlated - When one electron is close to the nucleus, the other tends to be far away - When one is on one side of a molecular orbital, the other tends to be on the other side• These effects are not considered in the HF method • A single Slater determinant also restricts us to one orbital configuration i.e. we fully define electron occupancy (not ideal in complex cases) • The general approach of describing the overall wavefunction of a molecule as a sum of Slater determinants is the basis of the configuration interaction (CI) method Configuration interaction (CI)• A solution to the problems with the HF approach is to use multiple Slater determinants (electron configurations) in our calculation Total wavefunction = c0+c1 π1π2π3π4 π1π2π3π4 e.g. c0= ,c1= π1π2π3π4½½ E.g. for He: In this example, c1 will be very small;the wavefunction is mainly 1s character • Crucially, considering more than one configuration (Slater determinant) provides information about electron correlation i.e. we recover some of the 1% missing in HF calculations.• Any configuration that isn't the ground state is, by definition, an excited state.• In these calculations we use excited state configurations to help gain information about the ground state• Adding more configurations (determinants) gives more information• We could consider configurations where we have moved only 1 electron (singly excited states) - configuration interaction of singles (CIS)• We could consider configurations where we have moved up to 2 electrons - configuration interaction of singles and doubles (CISD)• We could consider configurations where we have moved up to 3 electrons - configuration interaction of singles, doubles and triples (CISDT) • "Full CI" would correspond to all possible excitations• Full CI enables an exact solution of the Schrödinger equation (for a given set of orbital approximations) However, it is totally impractical in most cases due to the computational power needed • Slight alternatives give improvements to the CI method Multi-configurational self-consistent field method (MCSCF)• Assume excitations from low-energy occupied orbitals and to high-energy unnocupied orbitals unlikely to be significant One varient is “Complete active space SCF” (CASSCF):define “active” orbitals and carry out full CI within these Much less computationally expensive than real full CI,…but still complex Coupled Cluster (CC): another way of combining determinants • Essentially the same process as CI, but more computationally efficient• CC used much more commonly due to better efficiency• CCSD(T) (approximate treatment of triples) is a typical “gold-standard” choice Hartree Fock Method 1s, spin α1s, spinβ2s, spinα Li atom electron configuration: 1s2 2s1 3 spinorbitals assumed to combine to make the multi-electron wavefunction: • Mathematical description of this combination is the Slater determinant• Can work out the energy (via Fock operator)• Tweak orbital shapes to minimise energy (and get the best wavefunction)• What are we actually calculating?• Key aspect of HF method: each electron only sees average of all the otherelectrons (because of how the Fock operator works) sees an average of + , i.e. sees an average of sees an average of . = + + + i.e. i.e. i.e. We have important information missing, e.g.How does the motion of relate specifically to the motion of ? I.e. how are their motions correlated? The energy we calculate is missing theenergy from this type of e– e– interaction, called the correlation energy.How can we get around this with our limited description of electroninteractions?• Answer: take multiple ‘snapshots’ of averages and combine for more detail Post-HF methods are like taking multiple ‘noisy’ photos and combining themto get a clearer image. + + Neither original image has more detail than the other, but the combinedpicture is clearer. Post HF methods How do we make sure our snapshots are different? Force the electrons to moveKeep the nucleus and the number of electrons identical, but optimise awavefunction for multiple different orbital combinations, e.g.1s2 2s1 (ground state) 1s2 2s0 2p11s1 2s1 2p1 This gives 99 % of the energyi.e. almost all the detail These give information aboutelectron correlation (the 1 %) Remember, these orbitals are only shown offsetfor clarity! Variational methods Perturbation theory (alternative to variational methods) • Perturbation theory is used when a (complicated) problem differs only slightly from a (simple) problem already solved • Møller-Plesset perturbation theory does this for solving the Schrodinger equation - Complicated problem is solving the full Schrodinger equation - "Simple problem" is solving the Schrodinger equation without correlation effects (i.e. the HF method)Qualitatively, MP perturbation theory adds higher excitations to Hartree-Fock theory as a non-iterative correction • In HF theory: H0ψ0 = E0ψ0 Use Fock operator instead of H෡• Use Slater determinant to describe wavefunction• In perturbation theory: split Hamiltonian into two parts: H = H0+λV • H0 = HF Hamiltonian (Fock operator) • λ is an arbitrary real parameter that controls the size of the perturbation (0 < λ 1) • V is a small perturbation to the energy • MPn truncates at λn term: - MP1 = HF - MP2 corrects a significant amount of the correlation energy - MP3 not significantly better than MP2 - MP4 typically account for ca. 95% of correlation energy • MPn is not variational (unlike HF, CI, CC)• The energy obtained is not guaranteed to always be greater than the true energy. orderenergy HFMP2MP3MP4 (Approximate) Post-HF scaling (N is number of orbitals) Method Scaling HF N 4 MP2 N 5 MP3, CISD, CCSD N 6 MP4, CCSD N 7 MP5, CISDT N 8 MP6 N 9 MP7, CISDTQ, CCSDTQ N 10 FCI N ! Basis sets • We know what atomic orbitals should look like from exact solutions to the Schrödinger equation from the hydrogen atom • We need to be able to describe the AOs with some flexibility in our calculations • MOs are not exact combinations of AOs: • In chemistry a basis set is a group of mathematical functions used to represent thespinorbitals that combine to form the wavefunction • A basis set typically comprises sub-groups of functions that describe the shape ofthe constituent orbitals of a molecule Slater-type orbitals (STOs) • Replicate hydrogeinc orbitals well (solutions to the Schrodinger equation for the H atom) ψ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -6 -4 -2 0 2 4 6 atom 1 atom 2 atom 1 + atom 2 ψ • STOs are accurate, but integrals are hard to calculate (which is a crucial factor) socalculations are slow. • General uses: high accuracy atomic/diatomic systems Gaussian type orbitals (GTOs)• An alternative to STOs• Less accurate (don't replicate hydrogenic orbitals well) but fast to calculate 1s2p contracted Gaussian-type orbitals (CGTOs) • STOs give better accuracy• GTOs give better speed• Compromise: use combination of GTOs to approximate an STO• Slightly less accurate than real STOs, but faster• More accurate than single GTOs, but slower • STO-nG: STO function approximated using linear combination of n Gaussian functions STO-3G construction: 3 gaussians to replicate 1 Slater-type orbital: • STO-3G = single-ζ (zeta) or minimal basis set• Only one basis function for each orbital (but each basis function made of 3 Gaussians with a fixed ratio) – termed a CGTO• We could use multiple basis-functions for each orbital. If two are used, it is termed a double-ζ basis set• A triple-ζ basis-set describes each AO with 3 basis functions • More basis functions -> more flexibility -> better wavefunction description ->lower-energy (better) solution to the Schrödinger equation• But, more choice slows calculations Split-valence basis sets• Single-ζ basis sets (one basis function per AO)• A poor approximation for valence orbitals; bonding distorts valence orbitals• A good approximation for core orbitals; not involved in bonding so exhibit little distortion 3px(valence orbital)Involved in bonding so distortedFlexible basis functions needed2px(core orbital)Non-bonding so minimal distortionFlexible basis functions not requiredH2S • Split-valence basis sets: • Core orbitals described by single basis function • Valence orbitals described by multiple basis functions • Makes core orbital computations fast and valence orbital computations accurate Pople basis sets • Most commonly-used split-valence basis sets • Developed by (Nobel Laureate) John Pople • Standard in Gaussian software • n-ijG (double zeta) • n = number of Gaussian functions for each core (non-valence) orbital • i,j = numbers of Gaussian fucntions for each valence orbitals (2 values = double zeta) • E.g. 3-21G • Core orbital is a CGTO comprising 3 Gaussians • Valence described by two orbitals: 1 CGTO (2 Gaussians) and 1 single Gaussian • n-ijkG (triple zeta) • E.g 6-311G • Core orbital is CGTO comprising 6 Gaussians • Valence described by 3 orbitals: 1 CGTO (3 Gaussians) and 2 x single Gaussians. Example: 6-31G basis set for a C atom• Core orbitals: 1s• CGTO comprising 6 Gaussian functions• Valence orbital 2s• Described by 1 CGTO comprising 3 Gaussians and 1 single Gaussian• Valence orbital 2p• x, y and z each described by 1 CGTO comprising 3 Gaussians plus 1 single Gaussian 1s2s2px2py2pz 1 basis function: • 6 Gaussians 2 basis functions: • 3 Gaussians • 1 Gaussian 2 basis functions: • 3 Gaussians • 1 Gaussian 2 basis functions: • 3 Gaussians • 1 Gaussian 2 basis functions: • 3 Gaussians • 1 Gaussian This gives an exact expression to enable the energy of a system to be determined from the electron density (rather than the wavefunction) HF and DFT: key difference• Hartree Fock method is an approximate method that we solve exactly • Exact operator, but approximate wavefunction as Slater determinant• DFT is an exact method that we solve approximately • Approximate operator, but exact density • A specific DFT functional (method) is defined by how it treats this energy term• In practice, many functionals ignore the kinetic energy correction, and many also include empirical (from experiment) corrections thatnecessarily provide a correction for this VXC: what is it? Electron exchange: quantum mechanical interaction that arises from the indistinguishable nature of electrons and from the Pauli exclusion principleElectron correlation: the amount by which the movement of one electron isinfluenced by other electrons 𝐸𝑋𝐶 assumed to be dependent on density, ρ, at position, r• I.e. electron density at a given point defines 𝐸𝑋𝐶 exactly• This means that (separately) the contributions to the exchange and correlation energies are assumed to depend on the density at any given point in space Local Density Approximation (LDA) LDA gives good geometries but tends to underestimate exchange energy, and overestimate correlation Generalised Gradient Approximation (GGA)• Provides an extension to the local density approximation• Assumes 𝐸𝑋𝐶 is dependent on density, ρ, at position, r, as well as the gradient of ρ (i.e. how much the density is changing at that position) Atom 1Atom 2 Electron density Electron density Identical density, but different gradients,therefore different𝐸𝑋𝐶using generalisedgradient approximation Atom 1Atom 2 Electron density Electron density Identical density, therefore same 𝐸𝑋𝐶using local density approximation GGA represents a significant improvement over LDAAccuracy approaches that of methods such as MP2 and in some cases surpasses them A DFT method (or functional) usually comprises a choice of 1 method to define exchange, and another to define correlation Hybrid functionals• A common approach is to incorporate some exact exchange from HF theory• Use other sources to calculated the remainder of EXC • Typically the most common methods used in chemistry MethodError /kjmol-1HF649DFT (PBE)87DFT (BLYP)41DFT (B3LYP)40 DFT HF comparison DFT has same computational cost, far better results (generally) Both approximately scale as N4 (N= number of electrons) Limitations• DFT Calculations are not exact solutions of the full Schrodinger equation (exact functional unknown).• The Hohenberg-Kohn theorems apply only to ground state energies. Excited states can be predicted, but these predictions are not as good as ground-state predictions• DFT gives inaccurate results involving weak van der Waals attractions, which are a direct result of long range electron correlation (although corrections can be made) Single point energy• The simplest quantum chemical calculation• Determine total energy for fixed set of coordinates 𝐻ѱ= Eѱ• Decide on method: determines Hamiltonian approximation (i.e. accuracy)• E.g. HF, post-HF, DFT• Fixed atomic coordinates determine basis function origins• I.e. where the basis functions (atomic orbital representations) are in space• Use self-consistent-field (SCF) method to optimise ψ• Minimise energy for fixed coordinates• Relies on variational principle -1.12 -0.92 -0.72 -0.52 -0.32 -0.12 1 2 3 4 5 Energy / a.u. Iteration Improving wavefunctionLowering energy Finalwavefunction Initialguess Geometry optimisations• Probably the most common application of quantum methods• Determine lowest-energy atomic coordinates from initial guess• General method: • Perform SCF calculation to determine ψ and E for initial coordinate guesses • Displace coordinates slightly • Perform another SCF calculation to determine ψ and E• Repeat, e.g. until energy change is within a threshold • Local minima can be a big problem• Global minimum usually desired• No way of easily checking whether a calculation has found a local or global minimum 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 0 50 100 150 200 250 300 350 400 Relative E / kJ mol - 1 C - C - C - C dihedral / ° ButaneCH3-CH2-CH2-CH3 Frequency calculations • Absence of negative frequencies indicates a minimum-energy geometry• It does not indicate a global minimum; local minima will have only positive frequencies • Match with experimental frequencies provides confidence in calculated structures • 1 negative frequency indicates the structure is at a transition state (saddle-point) Potential energy scans• A method of studying energy changes with geometry modifications• Vary (and fix) 1 or more geometry constraints and calculate structural energies• Typical geometry modifcations• Atomic distance• Bond angle• Dihedral angle• Provides “slice” of potential energy surface for 1 variable 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 0 100 200 300 400 Relative E / kJ mol - 1 C - C - C - C dihedral / ° Solvated systems Adding explicit solvent molecules often adds huge amounts of complexity • Polarisable continuum model (PCM) commonly used• Average effect of solvent considered, rather than explicit molecules• Does not slow down calculations much Solvent continuum Molecular surface Atomic positions Quantum methods Classical methods Wave-description of electrons Standing waves in 2D Standing waves in 3D The wavefunction The Schrodinger equation H (Hamiltonian) is the sum of the kinetic and potential energy Solving equation for a given system should give acceptable wavefunctions and associated energies. "Particle in a box" solutionsA hypothetical experiment where an electron is constrained between 2 infinite energy barriers) 00aV x 0a 0a 0axxxΨ(x)n= 1n= 2n= 3 0a 0a 0axxx|Ψ(x)|2 Wavefunctions Probabilities 24 E Acceptable energies Energies in H atom are calculatedto be quantised Constants • ψ depends on multiple variables; we call these variables quantum numbers Quantum numbers can be defined as:“the sets of numerical values which give acceptable solutions to theSchrödinger wave equation for the hydrogen atom” Equations complex, but pictures are informative The Schrodinger equation cannot be solved (exactly) for more than 1 electron H atom solutions Variational Principle Visualisations Density functional theory (DFT): an alternative to the wavefunction HF and post-HF methods assume the quantum mechanical wavefunction, ψ, contains all the information about a given system Drawbacks:• Depends on a lot of variables: coordinates of each electron (x,y,z) and the spin state (up, down)• Extremely difficult to calculate exactly• Non-intuitive: gives the energy, but does not make much sense in itself Dependence of ψ on electron coordinates suggests that the electron density, ρ, may be a suitable alternative First Hohenburg-Kohn theorem• The electron density of any system determines all ground-state properties of the system• Suggests the density must define the external potential of the system, therefore the Hamiltonian, 𝐻, the wavefunction, ψ, and the energy of a system - Just as the wavefunction, ψ, defines all properties of a system, including the electron density, ψ2.• This theorem does not give any information as to how the density defines properties! Second Hohenburg-Kohn theorem• The correct ground state density for a system is the one that minimizes the total energy through the functional E[n(x,y,z)]• I.e. the variational principle applies • It is desirable to have an equivalent of the Hamiltonian in the Schrodinger equation that acts on the electron density to give an energy Starting point: assume that electrons do not interact non-classically Kinetic energyof non-interactingelectronsNuclear-electronpotential energyElectron-electronpotential energy 𝐾𝐸𝑛𝑖 [𝜌 (𝒓)] + 𝑉𝑛𝑒 [𝜌(𝒓)] + 𝑉𝑒𝑒[𝜌 (𝒓)] This provides an exact Hamiltonian for a system of electrons that only interact classically. Can solve it just like we do for the Schrodinger equation We still have a problem: we know electrons interact non-classically, and we know these interactions are significant• Solution = add corrections to our energy terms: 𝐾𝐸𝑛𝑖 [𝜌 (𝒓)] + 𝑉𝑛𝑒 [𝜌(𝒓)] + 𝑉𝑒𝑒[𝜌 (𝒓)] + Δ𝐾𝐸𝑛𝑖[𝜌 (𝒓)] + Δ𝑉𝑒𝑒 [𝜌 (𝒓)] Kinetic energyof non-interactingelectronsNuclear-electronpotential energyElectron-electronpotential energy Correction to electron-electron potential energydue to interactionCorrection to kineticenergy of electronsdue to interaction The correction is often abbreviated as 𝑉𝑋𝐶 or E𝑋𝐶 This energy term combines all non-classical (quantum) interactions Force fields Through-bond interactions Through-space interactions Bond stretching Angle bending Dihedral torsion Electrostatic interactions Lennard Jones interactions Sampling representitive configurations Monte Carlo MD MC comparison Calculation types Key concepts Anything collection of objects and positions may be described by a potential energy surfaceIn chemistry we consider collections of electrons, nuclei and atoms.The energy of a chemical system varies with the location of all the constituent particlesFull PES = energy known for every possible combination of variablesDimensions of PES scale with number of particles as 3N - 6(Translation and rotation of entire system not relevant to energy)Typically impossible to calculate full PES: too many combinations of variables Simplifying the PES Omit some areas Calculate route through PES Molecular energies Energies vs. populations (statistical thermodynamics) How do we use statistical thermodynamics?Some calculations do not need to use this relationship, e.g. in molecular dynamics the energies and populations are both calculatedSome calculations automatically include statistical thermodynamics, e.g. Metropolis Monte Carlo methodsSome calculations must have this releationship applied subsequently, e.g. if we want to know relative populations from a quantum method(which only gives us a system energy) AP0627 Theoretical andComputational Methods You can move across this overview using your mouse. You can scroll to zoom in and out or use '+' and '-' on your keyboard.Related items are contained within darker outlines, each of which has a heading. This should help you appreciate relationships between sections of the course that may not have been so obviousfrom lectures alone. Navigation This course overview is intended to help you see how different aspects of the course are related, and to give you an appreciation of the course as a whole. It is not an all encompassing summary or a replacement for the lectures and lecture notes. • Rarely used singly
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  1. AP0627 course overview